[plt-scheme] music theory (was Natural numbers)

From: Robby Findler (robby at eecs.northwestern.edu)
Date: Thu Mar 12 08:56:48 EDT 2009

Apropos of nothing, car horns seem to use the simpler rational
harmonies (not piano tunings), which I always enjoy. :)


On Thu, Mar 12, 2009 at 7:10 AM, Jos Koot <jos.koot at telefonica.net> wrote:
> Good Story. We are off topic I am afraid, but nevertheless the following:
> I am not sure that bach used the nowadays equally tempered scale for Das
> Whol Temperierte Klavier (I have been told differently once by a music
> teacher) But, as you point out, it certainly was not the mean-tone tuning.
> Mark though that a good singer, player of string or wind instruments (other
> than organ),orchestra, chorus or ensemble do not use the equally tempered
> scale all the time. In particular the terts of a final chord in major should
> be taken somewhat higher than in equally tempered scale in order to obtain
> good harmony.
> Jos
> ----- Original Message ----- From: "Stephen Bloch" <sbloch at adelphi.edu>
> To: "PLT Scheme ML" <plt-scheme at list.cs.brown.edu>
> Sent: Thursday, March 12, 2009 3:59 AM
> Subject: Re: [plt-scheme] music theory (was Natural numbers)
> snip
>> I'm not sure what you mean by "the multiplication wrinkle," but music
>> theorists have known that an octave corresponded to doubling or  halving of
>> something, and a fifth corresponded to a factor of 1.5, at  least since
>> Pythagoras.  If you recognize that an octave corresponds  to doubling or
>> halving, then a two-octave interval must correspond to  a factor of 4, a
>> three-octave interval to a factor of 8, and octave  and a fifth to a factor
>> of 3, and so on; again, I'm pretty sure this  has been known since
>> Pythagoras.  Of course, Pythagoras was measuring  string length or
>> wind-column length; the connection to frequency may  only go back to
>> Helmholtz in the 19th century (I'm not sure).
>> Anyway, this is enough to know that, as Jos points out, there is no
>> positive-integer number of perfect fifths which exactly matches any
>> positive-integer number of octaves, and therefore that Pythagorean  tuning
>> (tuning all the fifths to be exact 3:2 ratios, which is  actually pretty
>> easy to do aurally) can't possibly give you octave- invariant definitions
>> for all the notes of the scale.  In particular,  (3/2)^12 is very close to,
>> but not exactly, 2^7.  One solution is to  insert a "Pythagorean comma," a
>> fudge factor that makes one of the  fifths quite far off from a 3:2 ratio,
>> at some point in the scale;  ordinarily you put it someplace harmonically
>> remote, like from F# to  C#, someplace you'll never need if you're playing
>> in keys like C, F,  and G.
>> Pythagorean tuning was largely replaced in the 15th century (IIRC) by
>> "mean-tone" tuning, which also tries to get major thirds to be 5:4 ratios,
>> and minor thirds to be 6:5 ratios.  This turns out to be inconsistent BOTH
>> with perfect fifths AND with octave invariance, so  they inserted a couple
>> of other "commas" to fudge things into place.
>> What happened in Bach's time was the replacement of both Pythagorean  and
>> "mean-tone" tuning by "equal-tempered" tuning, in which each  semitone is
>> exactly a frequency ratio of the 12th root of 2.  Octaves  still correspond
>> to a 2:1 ratio, but NOTHING else is exactly any  integer ratio. The result
>> is that you can transpose to any key you  want and the piece will sound
>> pretty much the same, i.e. equally out- of-tune (from the perspective of
>> somebody accustomed to "perfect"  intervals).  Bach demonstrated the system
>> by writing a cycle of 96  short keyboard pieces, two preludes and two fugues
>> in each of the 12  major and 12 minor keys. He seems to have been
>> successful, because  almost all Western music written in the past 350 years
>> has been in  the equal-tempered system.
>> And now back to your usually-scheduled discussion of programming
>> languages....
>> Stephen Bloch
>> sbloch at adelphi.edu
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